Primal-dual block-proximal splitting for a class of non-convex problems

TL;DR

This paper introduces an advanced primal-dual block-proximal splitting algorithm tailored for non-convex, non-smooth optimization problems with block structure, achieving faster convergence through adaptive step lengths and acceleration, demonstrated on inverse problems.

Contribution

It extends primal-dual algorithms to non-convex problems with block structure, incorporating adaptive step lengths and acceleration for improved convergence rates.

Findings

Achieved local $O(1/N)$, $O(1/N^2)$, and linear convergence rates.

Demonstrated effectiveness on diffusion tensor imaging.

Validated performance on electrical impedance tomography.

Abstract

We develop block structure adapted primal-dual algorithms for non-convex non-smooth optimisation problems whose objectives can be written as compositions $G(x)+F(K(x))$ of non-smooth block-separable convex functions $G$ and $F$ with a non-linear Lipschitz-differentiable operator $K$. Our methods are refinements of the non-linear primal-dual proximal splitting method for such problems without the block structure, which itself is based on the primal-dual proximal splitting method of Chambolle and Pock for convex problems. We propose individual step length parameters and acceleration rules for each of the primal and dual blocks of the problem. This allows them to convergence faster by adapting to the structure of the problem. For the squared distance of the iterates to a critical point, we show local $O(1/N)$, $O(1/N^2)$ and linear rates under varying conditions and choices of the step lengths parameters. Finally, we demonstrate the performance of the methods on practical inverse problems: diffusion tensor imaging and electrical impedance tomography.